Finite Groups with G-Permutable Schmidt Subgroups

Abstract

Given a non-empty set X of a group G, a subgroup A is said to be X-permutable (respectively, X-h-permutable) with a subgroup B, if ABx = BxA, for some element x ∈ X (respectively, x ∈ X ∩ 〈A, B〉). A subgroup A of a group G is said to be X-permutable (respectively, X-hpermutable) in G if A is X-permutable (respectively, X-h-permutable) with every subgroup of G. In this paper, we study the structure of a finite group G with all its Schmidt subgroups G-permutable (respectively, Gh-permutable).